陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「Math 246A, Notes 4: singularities of holomorphic functions」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

Previous set of notes: Notes 3 . Next set of notes: Notes 5 .

In the previous set of notes we saw that functions that were holomorphic on an open set enjoyed a large number of useful properties, particularly if the domain was simply connected. In many situations, though, we need to consider functions that are only holomorphic (or even well-defined) on most of a domain , thus they are actually functions outside of some small singular set inside . (In this set of notes we only consider interior singularities; one can also discuss singular

已知结果和反例

Singularities come in varying levels of “badness” in complex analysis. The least harmful type of singularity is the removable singularity – a point which is an isolated singularity (i.e., an isolated point of the singular set ) where the function is undefined, but for which one can extend the function across the singularity in such a fashion that the function becomes holomorphic in a neighbourhood of the singularity. A typical example is that of the complex sinc function , wh

After removable singularities, the mildest form of singularity one can encounter is that of a pole – an isolated singularity such that can be factored as for some (known as the order of the pole), where has a removable singularity at (and is non-zero at once the singularity is removed). Such functions have already made a frequent appearance in previous notes, particularly the case of simple poles when . The behaviour near of function with a pole of order is well understood: f

证明或构造的主线

Unfortunately, there are isolated singularities that are neither removable or poles, and are known as essential singularities . A typical example is the function , which turns out to have an essential singularity at . The behaviour of such essential singularities is quite wild; 下面会 show here the Casorati-Weierstrass theorem , which shows that the image of near the essential singularity is dense in the complex plane, as well as the more difficult great Picard theorem which ass

Finally, there are the non-isolated singularities. Little can be said about these singularities in general (for instance, the residue theorem does not directly apply in the presence of such singularities), but certain types of non-isolated singularities are still relatively easy to understand. One particularly common example of such non-isolated singularity arises when trying to invert a non-injective function, such as the complex exponential or a power function , leading to

阅读时建议盯住的点

Suppose we are given a holomorphic function and a point in . For a sufficiently small radius , the circle and its interior both lie in , and the Cauchy integral formula tells us that

In addition, we have the Cauchy integral type formulae

值得单独记下的条目

• (ii) Conversely, if are complex numbers obeying the asymptotic bounds (5) , (6) , show that there exists a function holomorphic on the annulus obeying the Fourier expansion (7) and the inversion formula (8) .
• (ii) has a pole of order at for some if one has , and for all . Poles of order are known as simple poles , poles of order are double poles , and so forth.
• (iii) has an essential singularity if for infinitely many negative .
• (i) is closed in and discrete (i.e., all points in are isolated);
• (ii) is holomorphic on ; and
• (iii) Every is either a removable singularity or a pole of finite order.
• (a) If has a removable singularity at , and has a zero of order at once the singularity is removed, then .
• (b) If is holomorphic at , and has a zero of order at , then .

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源